HS-integral and Eisenstein integral mixed Cayley graphs over abelian groups

Abstract

A mixed graph is called second kind hermitian integral (or HS-integral) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called Eisenstein integral if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. Let Γ be an abelian group. We characterize the set S for which a mixed Cayley graph Cay(Γ,S) is HS-integral. We also show that a mixed Cayley graph is Eisenstein integral if and only if it is HS-integral.